Analysis of a nonuniform gas flow past a sphere

ANALYSIS OP A NONUN~PORM GAS FLOW PAST A SPHERE (~.~

~

a1~

n~Awmlm~

Po,~

~ )

PMM Voi.29, N°--l, 1965, pp.175-177 N.E. KHRAMOV (Moscow) (Received April 4, 1964)

We present some results of analysis for the nonuniform flow of a gas past a sphere using the method of Dorodnltsyn-Belotserkovskli [1 to 3]. i. Let there be a nonuniform supersonic flow directed against a sphere (Fig.l); the flow is symmetric with respect to the x-axls and is given as

w+=/l(O,r),

~=f~(O,r)

(z = - - r cos O, y = r s i n O )

(t.t)

Here w. is the magnitude of the velocity in the nonuniform flow, ~ is the inclination of the velocity with respect to ~ e a x l s of symmetry, f, and /a are continuous functions of the spherical coordinates 8 and r • It is required to determine the shape and position of the shock wave and the mixed flow in the region of influence. Let us refer the velocity w to the critical velocity a. , the density p te the stagnation 0 density of the unperturbed flow Po, the pressure p to p . a 3 , and the linear dimensions to the Fig. 1 radius o~ The sphere. Then, using the Bernoulli integral and the stream functions , , the system of equations of gas dynamics in spherical coordinates may be written [2] in the following form: a

0

0"-7-[r~ (P + pu 2) sin O] + - ~ - [r (our) sin 01 = r (2P + pv2)~in O

0 (r2TusinO) + ~(r~vsinO)=O, 0 Or g @ I

P--

2x

X -- I

1-- ~+t~

d,, dO --rP ( v ~dr- - r u

[ X + I h~(×-l)

p'

) sinO,

(t.2) ~=~[(~)

I/(l-x)

$

T, x =

i/(×-l)

t--z+l

w2

Here ~ = P / P × is the entropy function, and u and v are the velocity components along r and e . The unknown functions are u, v, ~, . . Let • denote the distance from the sphere to the shock along the ray e - const , we have the f o l l o w l n g e q u a t i o n : ds/d0=--(i+~) 191

~,, ( O + ~ )

(1.3)

192

N.E. Xhramov

The boundary conditions may be written a s follows: on the body r - 1 , u = O, ~ = O, tp = q~ (0) = const

(

2 q~(O)-- × + t

t {w+, w+Z~

on t h e s h o c k wave

(x'--l)( ~4---x--

r

.

i

x--I l--~w+'

)}(

t--

~ - - ~! w+ ' ) - l ) ×+

+ e(O),

u = w v sin 0 - - tv x cos 0,

ta x = w+ cos (~ - - b) cos ~ +

sin o w+ sin (~ -- 0) ~

v = wx sin 0 + wv cos O,

wv = w+ cos (z - - ~) sin ~ - -

~os w+ sin (* - - (~) ~

~, = [2.',.~- sin2 (~ -- O)

x--t(

'~=[ t -- ~~-iw +

×--t

1]

2 cos (z -- O) ]'

'c (1 -- ×z +--t I w +J ~1/¢~-1)

'=

Here e is the a~le of inclination o f t h e s h o c k wave t o t h e a x i s o f s y m m e t r y , and w= and w, a r e t h e v e l o c i t y c o m p o n e n t s a l o n g x and ~ . 2. L e t us c o n s i d e r as an e z ~ l e t h e f l o w f r o m a three-d30me;181c~lAt so~trce past a sphere. T~ree-dlmenslomLl source flow occurs outside a sphere of radlus r . • on w h i c h t h e v e l o c i t y r e a c h e s c r i t i c a l value, while the veloc i t y o u t s i d e t h e s p h e r e f o r a s u p e r s o n i c eotu~ce t n e r e a u s e s up t o ~f - - a t Infinity. L e t us p u t t h e s o u r c e a t a d l s t a n c e U from the origin In the negative direction of the x-axis (Fig.l). T~e n o n u n i f o r m f l o w J u s t a h e a d of the shock is given by

(I + 8) sin 8 0= 1 u'+ - - d

~,~-' C - - ( t + 8 )

cce0

1 ( 2 ~xt~x-1) sin ~ t) (! + e)* \ x - - + - ~ / V,w+sin'0

(1

:¢ ~ t --~w+j

,~--x/(x--x} (2.t)

Here d is the ratio of the sphere radius to r . . System (1.2}, (1.3) le solved by the method of intei~al relations. The region of Integration &S d£vided into strips equidistant in ~ . The integ r a n d functions a r e represented by interpolat2r~ polynomlals r with interpolating node points at the boundaries o£ the strips. For the nth approximation, the system of equations may be written schematically as in [2]

d~ dO - -

(! + e )

dt'i (' dO - - F i tt'i ° ' -

c o t ( o ~{- 0),

do

dut

dO - - F ,

dO

× _!_ ! _ '~ ~ - )i - l , . u x -- l ~ qi : % (~'i)

--Oi'

d vo dO - -

d* t ( dri _ dO -- riPi v i ~ (i = 2, 3 . . . . .

Fo l--t,0 ~

ritti )

sin 0

n)

(2.2)

H e r e F, ¥0, r t and t , a r e the . f u n c t i o n s d e f i n e d and holomorph~tc I n t h e region of integration. System (2.2) is integrated ntmerloe.lly ~the axls of symmetry 8 = O, where vt- v~- O, St= O, • - ~ , and the ~nlkalown paurameters u, and t h e Va~ue ¢ a r e aeteraLqZled f r o m t h e r ~ ~ t that the s o l u t i o n be re&-~tlar a t s ~ points. At e a c h s t e p Of t h ~ L ' l t ~ t ~ O ~ , we t&ke I n t o a c c o u n t t h e ~ t e r a Of t h e nonlxl.tforllflOW i n ~ 2 , 1 ) ~ ~he derivatives dO / dO and dw+ / dO i n t h e r l g h t - ~ e~des of (2.2). ~, We g i v e some r e s u l t s o f t h e n u m e r l c a £ c l ~ l e u l a t l o ~ , carrled the first approx.~tlon i n t h e r e g i o n o f aubsor~tc e~td ~ x ~ d f l o w s .

out in

Analysis

Let

or .Itn o n u n i f o r m g a s f l o w past a s p h e r e

193

N o denote the Mach number on the axis of symmetry Just ahead of the f

fy X= !

! Fig. 2

\ \\ \

\

d'=20

\ \ 42

~ ' d=l

V\, , \

\

),o' Q~

O.6

Pig. ~, for a uniform flow with

0

Pig. 3 s h o c k wave. I n F i g s . 2 and 3 a r e shown the shock waves and location of the sonic points on the sphere and shock for No" ~ and i0 . For M 0= ~ at d = ~ and 20, we have o = 2,28 and 1,27; for N o- I0 at ~ = 3, 50 and'lO0, we have o = IL~7, 3.15 and 1.39, respectively. To compare the cases, we show the shock waves obtained for the uniform flow past a sphere with M.- ~ and 10. Even a small nonuniformlty in the angle results in a significant shift of the sonic points towards smaller 0 and a decrease in the width of the shock layer. In all the cases studied, the sonic lines lie below the ray 0 - const passing through the sonic point on the sphere. In FIZ.~ is shown the pressure distribution p "P(O)/P(O) over the sphere for N 0 - - - - 1 at ~ different values of d The dashed curve shows the pressure

M.- ~ •

The author thanks his scientific co-worker I.I.Kuklin for carrying out the computations. n~LIOGP~APHY 1.

2.

3.

Dorodnltsyn, A.A., Ob odnom metode chlslenno8o reshenlla nekotorykh nelinelnykh zadach aerodlnamikl (On a Method of Numerical Solution of Some Nonllnear Aerodynamic Problems). Trudy vses.mat.S ezda, Akad.Nauk SSSR, Vol.3, 1958. ~ ! o t s e r k o v s k i l , O.M., 0 raschete obtekanlla oeesimmetrlchr~kh tel s otoshedshei udarnoi volnoi na elektronnoi schetnoi mashine (Calculat i o n o f f l o w p a s t a x i s ~ m n e t r i e b o d i e s w i t h d e t a c h e d s h o c k waves u s i n g an e l e c t r o n i c c o m p u t e r } . PMM V o l . 2 ~ , • 3, 1960. Belotserkovskil, O.M., S l m m e t r i c h n o e o b t e k a n l e z a t u ~ l e n n y k h t e l s v e r k h zvuRovym potokom s o v e r s h e n n o g o I r e a l ' n o g o . g a z a (Symmnetrlc s u p e r s o n i c i d e a l and r e a l g a s f l o w s p a s t b l u n t b o d i e s ) . Z ~ . v y c h i s l . Y a a t . m a t . F i z . VOI.2, NR 6, 1962. Translated by C.K.C.